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Probing Variations of the Rashba Spin-Orbit Coupling at the Nanometer Scale

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Probing Variations of the Rashba Spin-Orbit Coupling at the Nanometer Scale Probing variations of the Rashba spin-orbit coupling at the nanometer scale Jan Raphael Bindel1, Mike Pezzotta1, Jascha Ulrich2, Marcus Liebmann1, Eugene Sherman3, and Markus Morgenstern1 1II. Institute of Physics B and JARA-FIT, RWTH Aachen University, D-52074 Aachen, Germany 2 Institute for Quantum Information and JARA-FIT, RWTH Aachen University, D-52074 Aachen, Germany 3Department of Physical Chemistry, the University of the Basque Country UPV-EHU and IKERBASQUE, Basque Foundation for Science, Bilbao, Spain The Rashba effect as an electrically tunable spin-orbit interaction1 is the base for a multitude of possible applications2-4 such as spin filters3, spin transistors5,6, and quantum computing using Majorana states in nanowires7,8. Moreover, this interaction can determine the spin dephasing9 and antilocalization phenomena in two dimensions.10 However, the real space pattern of the Rashba parameter has never been probed, albeit it critically influences, e.g., the more robust spin transistors using the spin helix state6,11,12 and the otherwise forbidden electron backscattering in topologically protected channels.13,14 Here, we map this pattern down to nanometer length scales by measuring the spin splitting of the lowest Landau level using scanning tunnelling spectroscopy. We reveal strong fluctuations correlated with the local electrostatic potential for an InSb inversion layer with a large Rashba coefficient (~1 eVÅ). The novel type of Rashba field mapping enables a more comprehensive understanding of the critical fluctuations, which might be decisive towards robust semiconductor-based spintronic devices. The Rashba effect1, which lifts spin degeneracy by breaking inversion symmetry at surfaces or interfaces, was firstly probed in transport using the beating pattern in Shubnikov-de Haas oscillations15 or the weak antilocalization effect.16 Later, Rashba–split bands and their spin polarization were visualized by photoelectron spectroscopy.17 The first successful attempts to use the Rashba effect for spin manipulation required low temperatures and found relatively small signals,18-21 probably due to D’yakonov-Perel’-type spin randomizing by disorder. Options to overcome this limit are more one-dimensional or ballistic devices.21,22 Another method is to balance the Rashba and the Dresselhaus couplings,6 leading to a persistent spin helix with a momentum 1 independent spin rotation axis.11 For such cases, where the D’yakonov-Perel’ mechanism is suppressed, other dephasing mechanisms, such as fluctuations of the Rashba parameter, limit device functionalities. Interestingly, the topological protection in spin channels of 2D topological insulators13,14 is also likely limited by fluctuations of the Rashba parameter in combination with electron-electron interaction13 or spin impurities.14 A natural method of investigating electronic disorder is scanning tunnelling spectroscopy (STS).23-24 STS has already revealed fingerprints of the Rashba effect in two-dimensional electron systems (2DES) , such as a singularity at the band onset in the density of states (DOS)25,26, a beating pattern of the Landau levels in the DOS,27 or standing wave patterns from scattering between Rashba bands and other spin degenerate bands26, thereby circumventing the absence of quasiparticle interference between the two components of the Rashba-split band.28 However, these methods did not probe the spatial fluctuations of the Rashba effect. Here, we use an InSb 2DES, produced by Cs adsorbates on the (110) surface, to probe the Rashba parameter in real space. STS in magnetic field B reveals a nonlinear spin splitting of the Landau levels (LL), which fits to the Rashba model at intermediate B = 3-7 T, while exchange enhancement29,30 dominates at higher B. Thus, the spin splitting at intermediate B can be used to trace as a function of position R revealing that (R) spatially fluctuates between 0.4 eVÅ and 1.6 eVÅ. It exhibits a correlation length of 30 nm and a strong correlation with the electrostatic potential of the 2DES, as mapped as the spin-averaged LL energy. The sample is sketched in Fig. 1a. By adsorbing Cs on p-type InSb(110), the valence and conduction bands are bent down towards the surface such that an inversion layer with one occupied subband is formed (Fig.1b).27 The Cs coverage (1.8% of a monolayer) is low enough to allow a 2 mapping of the 2DES by STS.27,29,31 A strong electric field |푬| ≈ 3 ⋅ 107 V/m within the 2DES results from acceptor-doping,27 which in combination with the large atomic numbers of In and Sb leads to a large . Figure 1c shows the spin-split LLs of the 2DES according to the Bychkov- Rashba model.1 One recognizes crossing points of opposite spin levels (dashed ellipses) and a nonlinearity of the spin splitting at low B. Figure 1d shows this splitting for different , while keeping all other parameters identical. Different naturally lead to different nonlinearities, offering an elegant method to locally determine . Although is a strictly local parameter, the measured spin splitting is related to wave functions, such that the spatial resolution of the method is limited by about the cyclotron radius being smallest for the lowest level LL0. For smooth disorder potentials V(R) with respect to the magnetic length 푙B = √ℏ/푒퐵 (cyclotron radius of LL0), perturbation theory in 훁푉(푹)푙B/(ℏ휔푐) describes the energies n, adequately for different LLs n and spin labels = +,:32 ∗ 2 2 휆 푔푚 푉푛(푹)−푉푛−1(푹) 2√2훼(푹) 휀푛,휆(푹) = ℏ휔c (푛 − √(1 − + ) + 푛 ( ) ) + 2 2푚푒 ℏ휔c ℏ휔c푙B [1] 푉푛−1(푹)+푉푛(푹) 2 2 푉푛(푹) = ∫ 푉(푹 + 풓) ∙ 퐹푛(풓)푑 풓 [2] th ∗ Fn(r) is the kernel of the n LL wave function (supplement S1) and 휔c = 푒퐵/푚 is the cyclotron frequency. We can therefore determine (R) from the measured splitting 휀0,− − 휀1,+, for known V(R), 푔 and 푚∗. 3 Figure 2a shows the DOS, i.e. the spatially averaged local density of states (LDOS) of the 2DES at B = 7 T. The characteristic beating pattern of the LLs found previously27 can be used to estimate the average Rashba parameter 훼̅ by comparison with the different fitting lines. The best agreement with the experimental beating pattern is found for 훼̅ = 0.7 eV ∙ Å in agreement with 27 ∗ previous results. For simplicity, we used a constant 푚 = 0.03 ⋅ 푚푒 and 푔 = −21 (averaged value, see below), neglecting the band nonparabolicity for the lowest LL. This causes the discrepancies at higher LLs. The width of the LL peaks is directly taken from the distribution of V(R) (Fig. 3e). The observed strong dip at the Fermi level EF in the experiment is related to the well-known Coulomb gap.29,33 In order to extract the local Rashba parameter α(R), we recorded local LL fans. Figure 2b shows the measured LDOS of a single point at different energies and B. LLs and spin levels of two subbands are discernible as marked. The individual levels collectively undulate with B, which we ascribe to the undulation of all LLs with respect to EF in order to maintain the fixed carrier density and, to a lesser extent, to exchange enhancement.29 Reproducible instabilities in the spectroscopy are observed at distinct B (crosses, supplement Fig. S1c). Here, the conductance at EF partly drops to 3 pS, i.e. an insulating sample area close to integer filling factors prohibits current flow at these 29,33 B. We ascribe the slight suppression of LDOS around EF to a Coulomb gap. Multiple crossings of levels are present, e.g. in the boxes marked I-III enlarged in Fig. 2c. The dashed lines (guides to the eye) reveal that the marked levels cross away from B = 0 T, such that they cannot belong to simple Landau and spin energies, both being linear in B and crossing at B = 0 T. A natural way to explain the crossings is the Rashba effect and, indeed, some of the crossings appear at rather similar B as in the calculations of Fig. 1c. Discrepancies, most obvious at lower B, are attributed to the local confinement within the potential minimum, where the data 4 are recorded. This complication hampers the use of the crossings for an accurate determination of . Instead, we use the nonlinearity of the LL0 spin splitting, Δ퐸푆푆 = 휀0,−(퐵) − 휀1,+(퐵). Figure 2d shows the LDOS recorded at different positions at B = 14 T, exhibiting double peaks for LL0 and more complicated structures for higher LLs. The complex peak structures appear away 31 from the extrema of V(R) due the nodal structure of the LL wave functions. The splitting ESS determined from fitting two Lorentzians to the two peaks of LL0 is indicated. It increases if the peaks shift to higher energies, i.e., at higher V(R). This is opposite to the expectation from the nonparabolicity of 푔(푉), which decreases with increasing V. Furthermore, a fluctuating peak width is observed, which discussion is beyond the scope of this manuscript. Figure 2e shows Δ퐸푆푆(퐵) as deduced from Fig. 2b using Lorentzian fits (inset). Above B = 8 T, ESS oscillates exhibiting maxima at odd filling factors as expected for exchange enhancement.29,30,33 Since the exchange interaction depends exponentially on the overlap of the wave functions which roughly scales with 푙퐵 ∝ √1/퐵 , it decays rapidly at lower fields, being 29 below 1 meV for B < 6 T. Accordingly, oscillations of ESS are barely discernible at B < 7 T. Instead a largely linear ESS(B) is observed at 3-7 T decaying more rapidly at lower fields, similar to the curves of Fig. 1d. Extrapolating the linear part to B = 0 T (dashed line) reveals an offset of 0 31 Δ퐸푆푆~2.5 meV.
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